As explained for example in [1]},
the category of BPS states in type IIA string theory
on \({R}^{3,1}\times X\)  coincides with the bounded derived category \(\mathcal {C}=D^b(\operatorname{Coh}_c X)\)  of coherent sheaves on \(X\)  with compact support.
An object
\(E\in \mathcal {C}\)  is a complex of coherent sheaves \(\dots \rightarrow E^{-1} \rightarrow E^0 \rightarrow E^1 \rightarrow \dots \)  of arbitrary (but finite) length, where the component \(E^k\)  in cohomological degree \(k\)  is a coherent sheaf
with compact support on \(X\) .
We denote by \(\mathcal {H}^k(E)\)  the
cohomology of the complex at the \(k\) -th place,
and by \(\operatorname{ch}(E)\sum _k (-1)^k \operatorname{ch}(E^k)\)  the Chern character of the complex.
The homological shift \(E\mapsto E[1]\)  taking the complex \((E^k)_{k\in {Z}}\)  to \((E^{k-1})_{k\in {Z}}\)  maps D-branes with charge \(\gamma =\operatorname{ch}(E)=[r,d,\chi )\) 
to anti-D-branes with opposite charge \(-\gamma \) .
